Music to our ears: how the major scale began and works

Years ago, Dr. Issac Asimov was such a wonderful writer that he could explain anything so that it made perfect sense. Slide-rules, cholesterol, neutrinos, all were gist for his typewriter. In one memorable essay, "Music to my ears" he explained the design of the major scale so well that I still remember the principles 40 years after reading it. This posting is dedicated to the good doctor.

 

Picture this: you are in pre-medieval Europe, over a thousand years ago, one of a group of artisans  that are learning to craft and improve wonderful new inventions, musical instruments. You have newly invented tuning pegs and crude (but improving) soundboards so you can build lyras, harps and other members of the family of what I'll call string-board instruments. You start off slowly, with a muddy sound but are able to add more and more musically useful strings as your crafting skills improve.

But of course, there is a catch: no craftsman ever really works alone. You must craft your little marvels to placate multiple gods: what sounds good; what the local musicians can play; and alas, match what your other colleagues have crafted, so that they can be played together. Even in the pre-Renaissance past standards were enforced on craftsmen. 

The foundation of western music, the soul of a new instrument -

The Tonic and friends

So start with a string in the middle, tune it up, and call it the foundation, root or tonic note, then look for other tunings that go along with this tonic. The first and most obvious one is a string that is half as long, or twice as tight, or some combination of the two, that vibrates twice as fast. Call it the double or upper tonic.

The tonic and the upper tonic, plucked together sound smooth & consonant, yet different . Further, plucking just one makes the other string vibrate, unless muted by a finger, courtesy of the vibrations transmitted through the soundboard. You suddenly have variety: the number of possible sounds has leapt. From a toy, an instrument has emerged.

Relativity, thy name is music

Now in music, everything, but everything is relative: one just picks a spot, makes it the center and makes everything dance around it. A string can also vibrate half as fast as the tonic, at half the pitch. Relative to a "half-tonic" string the tonic is the upper-tonic, but since we're working relative to the tonic note, lets call it the Lower tonic.
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MajScale Dominant

So what else can we put into our instrument?  With a little experimentation you discover that a string tuned exactly 3/2 times sharper than the tonic sounds cool. Played against the tonic note it does not blend perfectly like the upper tonic, but neither does it fight. Instead it seems to dominate a bit over the tonic, remaining distinct, and perhaps feeling a bit louder.  Call this the Dominant.

And relativity still reigns. If there is above the tonic a note that is dominant to it, there must be a note below the tonic such that the tonic is dominant to. Call this the Subdominant. It lives up to it's name, sounding and feeling somewhat over-topped by the tonic. It is exactly is 2/3 the pitch of the tonic (2/3 * 3/2 = 1).

You and your fellow artisans continue the pattern. -

MajScale Relative Tonics

Filling in the dominant of the lower-Tonic (3/4 of the Tonic), and the sub-dominant of the upper-Tonic (4/3 of the Tonic). This makes for a nice little instrument, with no real dissonances and plenty of strings to pluck, mute or leave open. These three notes are still the foundation notes used by bass players everywhere.
The gap between the sub dominant and dominant notes all have a natural spacing that seems about right, so call it a Wholetone.

Musical Technology marches on.

Over time your instruments painstakingly improve. With these improved instruments, the notes last longer, are cleaner, purer and the strings hum together like they are alive, showing ever more subtle nuances and shading. With improved sound other special notes become clear.

In particular, halfway between the tonic and the Dominant is a spot just 5/4 higher than the tonic that sounds special. Since it is about midway between the tonic and the dominant, you call it the Mediant. -

MajScale mediant

Is there a Sub-Mediant? Of course(!), at the 4/5 point below the Tonic, perhaps a bit harder to discern.

So if you fill in the mediant for the root, dominant and subdominant you get a pretty nice set of notes, all clearly, closely related to the central tonic. Even the dullest audience member can hear it.

However, there a big gap between the tonic and the mediant. Fortunately there's another note you can add to fill it; the dominant of the dominant, this gives you a note between the tonic and the mediant, just above the tonic, the super-tonic. This gives you 3 sets of 3-note chords with a tonic-mediant-dominant structure, such a cool-sounding and useful chord that you call it the major chord. -

MajScale and more mediants

There are also three chords with a sub-mediant in them. These later chords have a different sound than the major chords, their sub-mediant note is buried under the Dominant and pushing a bit close to the Tonic, so sounds tight or restricted, so you naturally call them the Minor chords.
 
There's just one more note to name, the note just under the Tonic. Somehow, despite being created by playing only progressive sequences of notes it somehow has an implied dissonance against the tonic and the listener's ear knows it, even when the tonic is not playing. But this creates a cool musical tension that can make the music interesting. It creates notes that want to be "led back" to the tonic, creating a resolution. So call it the Leading Tone.-

MajScale Completing the Major Scale

 

Now you are really cooking, your instruments can make a huge combination of sounds. Even today over a thousand years later, if you can play the Three Chord Trick you have mastered the ability to play along with most common music .

This is instrument enough for several centuries of development of fine music. Enough time so that the notes are given a second set of names: that correspond to the number that they are from the tonic, in roman numerals; Arabic numerals and the concept of zero are still in the future:

 I      Tonic               Unity

 II     Super-Tonic      Second

III     Mediant            Third

IV     Sub-Dominant  Fourth

 V     Dominant         Fifth

VI     Sub-Mediant     Sixth

VII    Leading Tone    Seventh

VIII   (upper-)Tonic     Octave

With one or two exceptions, no new notes are named, all new notes are named relative to these named notes.

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Converting a harp into a keyboard

The instrument for those who can't play the harp

So you have a great set of instruments in these string-boards, what can you do to make them better? Well, casting around for a way to pluck the keys faster, some forgotten genius adds a mechanical string-plucker. Initially it's pretty simple, just a set of little boards that rocks like a teeter-todder with a "key" (a block of wood with a wooden finger that moves past the string) with a little piece of leather on the side of the finger that plucks the stings. Damn does the new Keyboard work! It becomes wildly successful; every concert-hall, every patrician villa, eventually every home must have one. Over time refinements like felt dampers, one-way plucking, felt hammers etc are added over a few centuries, evolving it in small steps into a piano.

The curse of success Standards can be a curse. OK, so you have this wonderful instrument, but you crave more (more to the point, customers are willing to pay for) innovations. But you are now shackled with the major scale and the fundamental design of your keyboard. What's to do? You add a new row or two of keys, of course.

Musical hop-scotch

There are multiple things that make sense to the musical ear. First, the tonic to dominant (I to V) relationship is musically huge, almost as strong as the Tonic relates to it upper or lower tonics so some musicians realize that they can innovate by playing a musical game of hopscotch, skipping from dominant to super dominant, "going up a Fifth". Of course, once this little game of "just one more dominant" begins, in four skips from the mediant (six from the root) you've hit a note very close to the Octave, you've "circled back" to the root in relative terms. This circling back to the tonic via a musical scenic route pattern or progression becomes known as the Circle of Fifths.

What about going the other way, going from the Sub-dominant to it's subdominant? This gives you a note a fourth below the Fourth that does not have a name (with one exception none of these fancy new notes has a name) called the Minor Seventh , halfway between the normal "major" Sixth and "major" Seventh. Again this note makes a whole pile of sense in some arrangements. Again, once this game starts you quickly fill in all the notes down to the octave below. and, of course, this gets called the Circle of Fourths.

Every keyboard needs more Mediants
This kind of simple musical arrangement using dominants was all the rage in the 8th and 9th centuries (a good link to Gregorian chant is wanted here, any suggestions?). But over time instruments improved in clarity and arrangers learned how to incorporate the mediant or third, into chords. But there's a wonderful ambiguity about thirds: should they be a major third above the root, or a major third below the Dominant? In the major scale some chords have mediants, the major third, and some are minor chords with a sub-mediant, a minor third in them. In the early days one could imagine a musician deftly retuning his or her instrument's thirds to suit the next song, but as the instruments grew heavier and more strings were added, this became a challenge.

So musician naturally wanted keys added the keyboards, so that every note had a mediant and a submediant.

Invasion of the "accidentals"
Musicial naming theory is an incredibly hoary and old thing. the black notes on the piano were added over 800 years ago, yet still have no real names, only relative names like Minor this or Augmented that, and are called accidentals.

More to come.
One day I'll write about the Tuning wars: "Just tuning" purists vs the "good enough" pragmatists.